I am thinking on the assumption that up until a certain age, parents have a right to refuse a DNA Test.
Also that you cannot simply go and snatch a child from the streets without proof. DNA obtained by stealth wouldn't do because there would be no chain of evidence.
I think you are on the right track as far as DNA is involved.
I think the police could ethically and legally carry out such tests in certain circumstances without permission being sought.
An example of DNA being obtained by police with cooperation happened quite recently in New Zealand ...
SnipThey investigated the first sighting and were "absolutely satisfied" it was not the missing girl but obtained a DNA sample after a request from Operation Grange, the British police task force investigating the disappearance.
Police said the New Zealand child's family provided the sample voluntarily and the DNA "does not provide a match for that of the missing girl".
https://www.telegraph.co.uk/news/newstopics/madeleinemccann/9866714/DNA-confirms-New-Zealand-girl-is-not-Madeleine-McCann.htmlAn example of DNA being checked without permission in the course of an investigation and with a more ambiguous result is ...
SnipThe DNA - taken from a drink bottle and straw after a girl was seen with a man at a cafe - matched that of a man, prosecutors in Tongeren town said.
It "doesn't mean that the presence of Maddie is excluded", they added.
http://news.bbc.co.uk/1/hi/uk/6936214.stmI think DNA evidence will play a major part in recovering Madeleine; the main problem I think my be gaining access to her to collect it.